Gundlagen der Künstlichen Intelligenz at TU München

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What is the performance of Inference by enumeration?

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Arc consistency of a variable 

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Define synatx


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Exemplary flashcards for Gundlagen der Künstlichen Intelligenz at the TU München on StudySmarter:

Initialization of the Arc consistency algorithm

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What simplifies a problem drastically?

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How can one retrieve a nearly tree-structured CSP? (2 options)

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Assume (A,B) was added to the the arc consistency queue. Which variable was just assigned a value and why?

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Define semantics

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Define model 

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Define Satisfaction

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Define entailment

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What is inference (Schlussfolgerung)? When can it be applied?

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Exemplary flashcards for Gundlagen der Künstlichen Intelligenz at the TU München on StudySmarter:

Gundlagen der Künstlichen Intelligenz

What is the performance of Inference by enumeration?

                                               

If KB and α contain n symbols, there are 2n models. Thus, the time complexity of enumeration is O(2n).

                   

The space complexity is only O(n) because the enumeration is depth-first.

                   

Later we show algorithms that are more efficient on average. However, propositional entailment is co-NP-complete, so every known inference algorithm is exponential in the size of the input.

                                       

Gundlagen der Künstlichen Intelligenz

Arc consistency of a variable 

                                                                       

Variable Xi is arc-consistent with variable Xj , if for every value in the domain Di there exists a value in Dj satisfying the binary constraint of the arc (Xi,Xj).

                       

Example: X is arc-consistent with Y for the constraint Y = X 2 if DX = {0,1,2,3} and DY = {0,1,2,3,4,5,6,7,8,9}, but Y is not arc-consistent with X (direction of the arc matters).

                                                           

Gundlagen der Künstlichen Intelligenz

Define synatx


                                                                       

Specifies how correct sentences are formed, e.g., x + y = 4 is well-formed, while x4y+ = is not.

                                                           

Gundlagen der Künstlichen Intelligenz

Initialization of the Arc consistency algorithm

                                               

  • after each assignment: after assigning variable Xi , add the arcs(Xj , Xi ) to queue, where Xj are all unassigned neighbors of Xi . 
  • as pre-processing: add all arcs of the CSP to queue.

                                       

Gundlagen der Künstlichen Intelligenz

What simplifies a problem drastically?

Identification of independent subproblems

-> can be solved in linear time

Gundlagen der Künstlichen Intelligenz

How can one retrieve a nearly tree-structured CSP? (2 options)

1. Conditioning

2. Tree decomposition

Gundlagen der Künstlichen Intelligenz

Assume (A,B) was added to the the arc consistency queue. Which variable was just assigned a value and why?

Variable B, because A all domain values of A have to be arc consistent with the domain values of b. The other way

Gundlagen der Künstlichen Intelligenz

Define semantics

                                                                       

The semantics defines the meaning of sentences, i.e., when a sentence is true. For instance, x + y = 4 is true for x = y = 2 and false for x = y = 1

                                                           

Gundlagen der Künstlichen Intelligenz

Define model 

                                                                       

Models are differently defined depending on the discipline. Here, models are instances which evaluate sentences to true or false. For instance, we have x men and y women playing a card game, then the sentence
x + y = 4 is true for the models x = 4, y = 0; x = 3, y = 1; and so on.

                                                           

Gundlagen der Künstlichen Intelligenz

Define Satisfaction

                                                                       

If a sentence α is true in model m, we say that m satisfies α. We use the notation M(α) to mean the set of all models of α.

                                                           

Gundlagen der Künstlichen Intelligenz

Define entailment

                                                         sentence a is a consequence of b

Entailment is the relationship between two sentences where the truth of one sentence requires the truth of the other sentence, which is written as

                       

α |= β
if α entails β. Formally, entailment is defined as

                       

α |= β if and only if M(α) ⊆ M(β). For instance, the sentence x = 0 entails xy = 0.

                                                           

Gundlagen der Künstlichen Intelligenz

What is inference (Schlussfolgerung)? When can it be applied?

Act or process of deriving logical conclusions from known premises.

                                                                     

1. After each assignment: see Inference() in backtracking algorithm. 

2. As pre-processing: before applying the backtracking algorithm.

                                                           

                                                           

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